On the Coarse Geometry of James Spaces

In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space J nor into its dual J*. It is a particular case of a more general result on the nonequi-coarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic structure. This allows us to exhibit a coarse invariant for Banach spaces, namely the non-equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property Q of Kalton. We conclude with a remark on the coarse geometry of the James tree space JT and of its predual.